Elementary Statistics

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Elementary Statistics

• Sampling distribution v.s Confidence interval

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Where are we?

• Scientific Method
• Formulate a theory -- hypothesis (Chapter1)
• Collect data to test the theory-sampling method,
  and experimental design (Chapter 2, Chapter3)
• Analyze the results --- graphically, numerically
  (Chapter 4, 5)
• Use models Chapter 6
• Understand the language of probability Chapter 7
• Interpret the results and make a decision
  --p-value approach
• Make a decision on proportion.- Chapter 8,9

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Confidence Interval Estimation

• Point estimate use the sample proportion to
  estimate the population proportion
• Confidence interval estimate - estimate the
  population proportion p is in an interval of
  value
• Confidence level - the probability that the
  interval contains the parameter p.

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68-95-99.7 rule
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Standard Error Revisited

• Standard Error
• approximate 95 confidence interval for is given
  by

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P is unknown Standard Error
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Study Chronic Fatigue Common

• A study was performed to learn about the
  proportion of adults who suffer from chronic
  fatigue syndrome. For this study, 4000 randomly
  selected members of a health maintenance
  organization in Seattle were surveyed for the
  illness. Surveys were mailed asking respondents
  if they felt unusual fatigue that interfered with
  work or responsibilities at home for at least six
  months. Of the 3066 people who responded
  (possible non-response bias), 590 reported
  chronic fatigue.
• We wish to estimate the proportion of adults who
  think they suffer from chronic fatigue syndrome.

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If we repeated this procedure over and over,
yielding many 95 confidence intervals for , we
would expect that approximately 95 of these
intervals would contain and approximately 5
would not.
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Other confidence Level
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Calculator-abbreviated 1-PropZint
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Use calculator for confidence interval

• 1-PropZInt
• X --- number of successes
• N --- sample size
• C-level --- confidence interval
• Click calculate

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Short Examples

• USA today conducted a poll based on telephone
  interviews with 1538 National Adults( From
  October 22-24)
• 51 of those will vote for Bush/Cheney
• Please explain For results based on the total
  sample of National Adults, one can say with 95
  confidence that the margin of sampling error is
  3 percentage points.

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Let's Do It! 9.8 9.8 What Is the Estimate?

• The 99 confidence interval for was calculated
  to be (0.27 , 0.42).
• (a) What is the value for the sample proportion
  ?
• (b) What is the value of the margin of error?
• (c) Give two suggestions for how you might
  reduce the margin of error.

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Where are we?

• Scientific Method
• Formulate a theory -- hypothesis (Chapter1)
• Collect data to test the theory-sampling method,
  and experimental design (Chapter 2, Chapter3)
• Analyze the results --- graphically, numerically
  (Chapter 4, 5)
• Use models Chapter 6
• Understand the language of probability Chapter 7
• Interpret the results and make a decision
  --p-value approach
• Make a decision on proportion.- Chapter 8,9
• Make a decision on population mean Chapter 8,10

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Make decision based on a sample mean

• For example
• Statement The national average gas price is
  2.00 per gallon
• Randomly picked 300 gas station show an average
  1.90 per gallon.
• Q Should we accept the statement?

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Sample distribution of sample mean

• The sampling distribution of the sample mean is
  the distribution of values of the sample mean in
  all possible random samples of the same size n
  taken from the same population.
• What does mean of sample tell you the mean of the
  population?

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Review
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Example

• If it was claimed that the average height of baby
  boy at birth is 19.75 inch, with standard
  deviation 0.75 inch.
• A random sample of 100 baby boy was select, what
  is the sampling distribution of the sample mean?

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Example 8.9 Sketch the Distribution of the Sample
Mean
X filling weight of bins, has a normal
distribution with a mean 300 pounds and a
standard deviation 5 pounds. A simple random
sample of 25 bins will be obtained, and the
sample mean filling weight will be computed.
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Lets Do It! 8.9 Probability of Accepting the
Shipment
The model for breaking strength of steel bars is
normal with a mean of 260 pounds per square inch
and a standard deviation of 20 pounds per square
inch. What is the probability that a randomly
selected steel bar will have a breaking strength
greater than 250 pounds per square
inch? 0.6915 A shipment of steel bars will be
accepted if the sample mean breaking strength of
a random sample of 10 steel bars is greater than
250 pounds per square inch. What is the
probability that a shipment will be
accepted? 0.9429
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Hypothesis about sample mean